Characterization of fluids with drag reducing additives in a couette device

ABSTRACT

A method is provided for characterizing fluid flow in a pipe where the fluid includes a drag reducing polymer of a particular type and particular concentration. A computational model is configured to model flow of a fluid in a pipe. The computational model utilizes an empirical parameter for a drag reducing polymer of the particular type and the particular concentration. The computational model can be used to derive information that characterizes the flow of the fluid in the pipe. The empirical parameter for the particular type and the particular concentration of the drag reducing polymer can be identified by solving another computational model that is configured to model turbulent Couette flow in a Couette device for a fluid that includes a drag reducing polymer of the particular type and the particular concentration. The empirical data needed for identification of the empirical parameter are obtained from Couette device experiments.

BACKGROUND Field

The present application relates to laboratory analysis of fluids,particularly fluids with drag reducing additives added thereto.

Related Art

Drag reducers are chemical additives, which being added to a fluid,significantly reduce friction pressure losses on fluid transport in aturbulent regime through pipelines. Such chemical additives, usuallypolymers, may decrease pressure drop by up to 80 percent, and thusallows reducing the friction losses to the same extent.

The efficiency of drag reducers is usually tested in a flow loop. For agiven drag reducer type and concentration a pressure drop along alaboratory pipe is measured at the Reynolds number that is maximallyclose to that expected in an industrial pipeline. A relative reductionin the pressure drop, in comparison to that in a flow free of dragreducers, is a measure of the additive efficiency.

Kalashnikov, V. N., “Dynamical Similarity and Dimensionless Relationsfor Turbulent Drag Reduction by Polymer Additives,” Journal ofNon-Newtonian Fluid Mechanics, Vol. 75, 1998, pp. 1209-1230, describes aTaylor-Couette device used for studies of turbulent drag reductioncaused by polymer additives. The Taylor-Couette device includes arotating outer cylinder and an immobile inner cylinder. An effect of adrag reducer was evaluated by the torque, applied to the inner cylinder.The greater reduction in torque resulting from the additive resulted inimproved drag reducer performance. The drag reduction was investigatedfor a wide range of Reynolds numbers and the author suggestsdimensionless criteria for drag reduction characterization.

Koeltzsch et al., “Drag Reduction Using Surfactants in a RotatingCylinder Geometry,” Experiments in Fluids, Vol. 24, 2003, pp. 515-530,studies turbulent drag reduction in a device of a similar design. Notethat measurement of the torque applied to the inner cylinder has alimited accuracy due to unavoidable friction in bearings.

SUMMARY

The present application provides a method of characterizing fluid flowin a pipe where the fluid includes a drag reducing polymer of aparticular type and particular concentration. A computational model isconfigured to model flow of a fluid in a pipe. The computational modelutilizes an empirical parameter for a drag reducing polymer of theparticular type and the particular concentration. The computationalmodel can be used to derive information that characterizes the flow ofthe fluid in the pipe.

In one embodiment, the empirical parameter for the particular type andconcentration of the drag reducing polymer can be derived by solvinganother computational model that is configured to model turbulent flowin a Couette device for a fluid that includes a drag reducing polymer ofthe particular type and concentration. The solution of the empiricalparameter for the particular type and concentration of the drag reducingpolymer can calculated from experimental data derived from operation ofthe Couette device with a fluid that includes a drag reducing polymer ofthe particular type and concentration.

In another embodiment, the computational model of the pipe flow includesa drag reduction parameter that is a function of the empiricalparameter. The drag reduction parameter is a function of a dimensionlesspipe radius R⁺. For example, the computational model of the pipe flowcan be configured to relate the drag reduction parameter to theempirical parameter by an equation of the form:D _(*)=1+α_(*) R ⁺

where

-   -   D_(*) is the drag reduction parameter,    -   α_(*) is the empirical parameter, and    -   R⁺ is the dimensionless pipe radius.

In yet another embodiment, the computational model of the pipe flowincludes a friction factor that is a function of the drag reductionparameter, wherein the friction factor relates pressure loss due tofriction along a given length of pipe to the mean flow velocity throughthe pipe. For example, the computational model of the pipe flow can beconfigured to relate the friction factor to the drag reduction parameterby an equation of the form:

$\frac{1}{f^{0.5}} = {{4{\log_{10}\left( {{Re}\; f^{0.5}} \right)}} + {8.2D_{*}^{2}} - 8.6 - {12.2\log_{10}D_{*}}}$

where

-   -   f is the friction factor,    -   D_(*) is the drag reduction parameter, and    -   Re is the Reynolds number of the flow in the pipe.        The computational model of the pipe flow can be further        configured to relate the Reynolds number Re to a dimensionless        pipe radius R⁺. The information derived from the computational        model of the pipe flow can include a solution for the friction        factor f for given flow conditions and possibly a pressure drop        over a given length of pipe based on the solution for the        friction factor f.

The computational model of the pipe flow can be based upon arepresentation of the flow as two layers consisting of a viscous outersublayer that surrounds a turbulent core.

The computational model for the turbulent Couette flow can be based upona representation of the turbulent Couette flow as three layersconsisting of viscous outer and inner sublayers with a turbulent coretherebetween.

In one embodiment, the Couette device defines an annulus between firstand second annular surfaces, and the computational model for theturbulent Couette flow includes a first drag reduction parameterassociated with the first annular surface and a second drag reductionparameter associated with the second annular surface, wherein both thefirst and second drag reduction parameters are also functions of theempirical parameter specific to a drag reducing polymer of theparticular type and the particular concentration. The first and seconddrag reduction parameters are also functions of a dimensionless torque Gapplied to the Couette device rotor.

The computational model for the turbulent Couette flow can also be basedon an equation that defines a fluid velocity at a boundary of a viscoussublayer adjacent one of the first and second annular surfaces. Suchequation can be derived by momentum conservation for a turbulent core.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic cross-sectional diagram of an exemplarycylindrical Couette device.

FIG. 2 is a schematic diagram of the annular surfaces of the cylindricalCouette device of FIG. 1.

FIG. 3 is a schematic diagram of a sensing apparatus that employs theLenterra technique to measure shear stress of the fluid at the surfaceof the wall of the outer cylinder of the cylindrical Couette device ofFIG. 1.

FIG. 4 is a schematic diagram of the layers of the flow field used in acomputational model of turbulent Couette fluid flow in the cylindricalCouette device of FIG. 1.

DETAILED DESCRIPTION

FIG. 1 shows an exemplary cylindrical Couette device 100, which includesa top wall 102, outer cylinder 104, inner cylinder 120, and bottom wall106 that define the boundary of an annulus 108 disposed between theinner surface 104A of the outer cylinder 104 and the outer surface 120Aof the inner cylinder 120. The Couette device 100 further has top andbottom retaining plates 110, 112 set apart by spacers 114 andmechanically secured, for example, by nuts and bolts 116. A fluid isloaded into the annulus 108 preferably through one or more fluid paths(e.g., one shown as port 119 and passageway 118).

The inner cylinder 120 is mounted on bearings and is coaxial with theouter cylinder 104. The outer cylinder 104 is fixed in position and thusremains stationary. The inner cylinder 120 rotates independently of theouter cylinder 104. A shaft 122 extends down from the bottom of theinner cylinder 120. A motor 124 has an output shaft 124A that ismechanically coupled to the shaft 122 by means of a coupling device 128,which can be a magnetic coupler, a rigid coupler, a flexible coupler, orother suitable coupling mechanism. In the preferred embodiment, themotor 124 can operate over a wide range of rotational speeds (e.g.,100-20,000 rpm) for rotating the inner cylinder 120 at different angularvelocities.

Instrumentation can be added to the Couette device 100 as needed. Forexample, devices for heating and/or cooling the fluids within theannulus 108 of the Couette device 100 may be added. Such devices may beused in conjunction with loading fluid into the annulus 108 to achieve apredetermined pressure in the annulus 108. Pumps are used to transferthe fluids into the annulus 108. The pumps define and maintain thepressure of the system. One or more temperature sensors and one or morepressure sensors can be mounted adjacent the annulus 108 to measurefluid temperature and pressure therein. In one embodiment, therotational speed of the inner cylinder 120 is measured through the useof a proximity sensor, which measures the rotational speed of the shaft122 mechanically coupled to the inner cylinder 120.

A schematic diagram of the Couette device 100 is shown in FIG. 2, withthe radius R denoting the radius of the inner wall surface 104A of theouter cylinder 104 and the radius r₀ denoting the radius of the outersurface 120A of the inner cylinder. The annulus or gap 108 between theinner and outer cylinders has a width H of (R−r₀) and a height of L. Afluid mixture that employs a drag reducer of a specific type andconcentration is loaded into the annulus 108 via port 119 and passageway118. The motor 124 is operated at a sufficient speed to provideturbulent flow of the fluid in the annulus 108 such that the Reynoldsnumber of the Couette flow Re_(c) exceeds 1.3×10⁴. The Couette flowcirculates in the annulus 108 during such operations.

The shear stress of the fluid at the inner surface 104A of the wall ofthe outer cylinder 104 is measured using the Lenterra technique thatcombines a floating element 301 and a mechanical cantilever beam 303with a micro-optical strain gauge (fiber Bragg grating or FBG) 305 asshown in FIG. 3. The shear stress is applied to the floating element 301attached to the cantilever beam 303. The floating element is installedflush with the inner wall surface 104A of the outer cylinder 104 in asensor enclosure 307. Displacement of the floating element 301 leads tobending of cantilever beam 303. When the cantilever beam bends, the FBGis strained in a manner that shifts its optical spectrum. Byinterrogating the FBG with a light source via optical fibers 309, thisstrain (and therefore the shear stress) is measured by tracking theshift in the resonant wavelength. The shear stress is calculated asτ_(w)=kΔλ, where k is the calibration coefficient and Δλ is the shift inthe resonant wavelengths. This technique provides more accurate shearstress measurements (as compared to the measurement of the torqueapplied to the Couette device spindle as is commonplace in many Couettedevices). The shear stress measuring sensor devices of FIG. 3 areavailable from Lenterra, Inc. of Newark, N.J., USA.

A. Couette Device Computational Model

The Couette fluid flow in the annulus 108 of the Couette device 100 canbe studied in terms of the dimensionless torque G and the Reynoldsnumber Re_(c) for such fluid flow. The dimensionless torque G is definedas a function of the torque T derived from shear stress τ_(w) measuredat the inner wall surface 104A of the outer cylinder 104 of the Couettedevice 100 as follows:

$\begin{matrix}{G = \frac{T}{\rho\; v^{2}L}} & (1)\end{matrix}$

-   -   where T=τ_(w)2πR²L, R is the radius of the inner wall of the        outer cylinder of the Couette device, ρ is the density of the        fluid, ν is the kinematic viscosity of the fluid, and L is the        height of the gap of the Couette device.

The Reynolds number Re_(c) for Couette fluid flow in the annulus 108 ofthe Couette device 100 can be calculated as:

$\begin{matrix}{{Re}_{c} = \frac{\omega\;{r_{0}\left( {R - r_{0}} \right)}}{v}} & (2)\end{matrix}$

-   -   where r₀ is the outer radius of the inner cylinder 120 of the        Couette device 100, and        -   ω is the rotor angular velocity of the inner cylinder 120 of            the Couette device 100.

To model the Couette flow in the Couette device 100, the flow field ofturbulent Couette fluid flow in the Couette device 100 can be describedby three layers including a relatively viscous outer sublayer 151adjacent the inner surface 104A of the outer cylinder 104, a viscousinner sublayer 155 adjacent the outer surface 120A of the inner cylinder120, and a turbulent layer or core 153 between the viscous inner andouter sublayers 151, 155 as shown in FIG. 4. The streamwise (tangential)velocity of the fluid flow in the Couette device 100 is shown as vectoru in FIG. 4. Velocity fluctuations in the fluid flow can have astreamwise component u′ and a radial component v′ as shown.

The model assumes a linear velocity distribution across the viscousouter sublayer 151:u ⁺ =y ⁺ ,y ⁺≤δ₀ ⁺  (3)

-   -   where u⁺ is the normalized fluid velocity given by u⁺=u/u_(0*)        where u is the streamwise velocity of the viscous outer sublayer        151 and u_(0*) is the friction velocity of the viscous outer        sublayer 151 given by u_(0*)=(τ_(w)/τ)^(0.5),        -   y⁺ is given as y⁺=u_(0*)y/ν where y is the distance from the            inner wall 104A of the outer cylinder 104, and        -   δ₀ ⁺ is the dimensionless thickness of the viscous outer            sublayer 151, δ₀ ⁺ is set to a predetermined value such as            11.6.    -   The velocity distribution across the turbulent core 153 for the        region confined by the boundary of the viscous outer sublayer        151 and the gap centerline 154 at R_(m)=0.5(r₀+R) is described        by the ordinary differential equation as:

$\begin{matrix}{\frac{d\frac{u}{r}}{dr} = {- \frac{u_{0^{*}}R}{{\kappa\left( {R - r} \right)}r^{2}}}} & (4)\end{matrix}$

-   -   where κ is the Von Karman constant, which can be set to a        predetermined value such as 0.45; other suitable values of the        Von Karman constant can be used; note that a decrease in the Von        Karman constant can require a corresponding reduction in δ₀ ⁺,        while an increase in the Von Karman constant can require a        corresponding increase in δ₀ ⁺.

The initial condition for Eq. (4) is the normalized velocity at theviscous outer sublayer surface boundary u⁺ (δ₀ ⁺) equal to a parameter λ(i.e., u⁺(δ₀ ⁺)=λ). In one embodiment, the parameter λ is set to apredetermined value such as 11.6 assuming the dimensionless velocitydistribution across the laminar sublayer in a Couette flow is identicalto that in the pipe wall vicinity.

The analytical solution of Eq. (4) is given by:

$\begin{matrix}{\frac{u(r)}{u_{0^{*}}} = {{\frac{1}{\kappa}\left( {1 + {\frac{r}{R}{\ln\left( {\frac{R}{r} - 1} \right)}}} \right)} + {\gamma\frac{r}{R}}}} & (5) \\{where} & \; \\{{\gamma = \frac{\lambda - {\frac{1}{\kappa}\left( {1 + {\left( {1 - a} \right){\ln\left( \frac{a}{1 - a} \right)}}} \right)}}{1 - a}};} & (6)\end{matrix}$

-   -   where α=δ₀ ⁺/R⁺ where R⁺ is the dimensionless radius of the        inner wall surface 104A of the outer cylinder 104 of the Couette        device 100.

R⁺ can be expressed through the dimensionless torque G as follows:

$\begin{matrix}{R^{+} = \sqrt{\frac{G}{2\pi}}} & (7)\end{matrix}$

The momentum conservation equation for the turbulent core 153 for theregion confined by the gap centerline 154 at R_(m)=0.5(r₀+R) and theouter surface of the viscous inner sublayer 155 can be given as:

$\begin{matrix}{\frac{d\frac{u}{r}}{dr} = {- \frac{u_{0^{*}}R}{{\kappa\left( {r - r_{0}} \right)}r^{2}}}} & (8)\end{matrix}$

The initial condition for Eq. (8) can derived from the normalizedvelocity at the gap centerline 154 at R_(m)=0.5(r₀+R) according to Eq.(5) as follows:

$\begin{matrix}{\frac{u\left( R_{m} \right)}{u_{0^{*}}} = {{\frac{1}{\kappa}\left( {1 + {\frac{1 + \eta}{2}{\ln\left( \frac{1 - \eta}{1 + \eta} \right)}}} \right)} + {\gamma\frac{1 + \eta}{2}}}} & (9)\end{matrix}$

-   -   where η is the ratio r₀/R.

Then, the analytical solution of Eq. (8) is given by:

$\begin{matrix}{\frac{u(r)}{u_{0^{*}}} = {{\frac{u\left( R_{m} \right)}{u_{0^{*}}}\frac{r}{R_{m}}} + {\frac{1}{\kappa}\frac{R}{r_{0}}\left( {{- 1} + \frac{r}{R_{m}} + {\frac{r}{r_{0}}{\ln\left( \frac{1 - \frac{r_{0}}{R_{m}}}{1 - \frac{r_{0}}{r}} \right)}}} \right)}}} & (10)\end{matrix}$

The circumferential velocity U_(i) is the velocity of the rotatingcylinder surface 120A of the inner cylinder 120 and calculated as:U _(i) =ωr ₀.  (11a)The circumferential velocity U_(i) can also be calculated by:U _(i) =u(r ₀+δ_(i))+λu _(i*)  (11b)

-   -   where δ_(i) is the thickness of the viscous inner sublayer 155        adjacent the outer surface 120A of the inner cylinder 120,        -   u is the streamwise velocity for the viscous inner sublayer            155,        -   λ is a parameter set to a predetermined value such as 11.6,            and        -   u_(i*) is the friction velocity of the viscous inner            sublayer 155 at the outer surface 120A of the inner cylinder            120.

Eq. (11b) can be rewritten as follows:

$\begin{matrix}{\frac{U_{i}}{u_{0^{*}}} = {\frac{u\left( {r_{0} + \delta_{i}} \right)}{u_{0^{*}}} + \frac{\lambda}{\eta}}} & (12)\end{matrix}$

The left-hand side of Eq. (12) can be expressed through thedimensionless torque G and the Reynolds number Re_(c) to obtain:

$\begin{matrix}{{\frac{\left( {2\pi} \right)^{0.5}}{1 - \eta}\frac{{Re}_{c}}{G^{0.5}}} = {\frac{u\left( {r_{0} + \delta_{i}} \right)}{u_{0^{*}}} + \frac{\lambda}{\eta}}} & (13)\end{matrix}$

For the right-hand side of Eq. (13), the velocity u (r₀+δ_(i)) can beequated to u (r₀+δ₀η) and then calculated by Eq. (10) to give:

$\begin{matrix}{\frac{u\left( {r_{0} + \delta_{i}} \right)}{u_{0^{*}}} = {{\frac{u\left( R_{m} \right)}{u_{0^{*}}}\frac{2\;\eta}{1 + \eta}\left( {1 + a} \right)} + {\frac{1}{\kappa}\left( {{- \frac{1}{\eta}} + \frac{2\left( {1 + a} \right)}{1 + \eta} + {\frac{1 + a}{\eta}\ln\frac{\left( {1 - \eta} \right)}{\left( {1 + \eta} \right)}\frac{\left( {1 + a} \right)}{a}}} \right)}}} & (14)\end{matrix}$

Eqs. (9), (13) and (14) represent a computation model for Couette flowwithout a drag reducer that can be solved to calculate the relationshipof the dimensionless torque G as a function of the Reynolds numberRe_(e) for the Couette flow without a drag reducer.

B. Extension of Couette Device Computational Model to Account for DragReducer

The computational model for the Couette flow without a drag reducer asdescribed above can be extended by considering two distinct dragreduction parameters: the drag reduction parameter D_(0*) for theviscous outer sublayer 151, and the drag reduction parameter D_(i*) forthe viscous inner sublayer 155.

The drag reduction parameter D_(0*) for the viscous outer sublayer 151can be related to the parameter α_(*) that is a function of the dragreducer agent type and its concentration as follows:

$\begin{matrix}{D_{0^{*}} = {1 + {\alpha_{*}\frac{u_{0^{*}}H}{{2\; v}\;}}}} & \left( {15a} \right)\end{matrix}$

where

-   -   u_(0*) is the friction velocity of the viscous outer sublayer        151 given by u_(0*)=(τ_(w)/ρ)^(0.5),    -   H is the gap width of the Couette device (H=r₀−R), and    -   ν is the kinematic viscosity of the fluid.

Similarly, the drag reduction parameter D_(i*) for the viscous innersublayer 155 can be related to the parameter α_(*) that is a function ofthe drag reducer agent type and its concentration as follows:

$\begin{matrix}{D_{i^{*}} = {1 + {\alpha_{*}\frac{u_{i^{*}}H}{{2\; v}\;}}}} & \left( {15b} \right)\end{matrix}$

where

-   -   u_(i*) is the friction velocity of the viscous inner sublayer        155,    -   H is the gap width of the Couette device (H=r₀−R), and    -   ν is the kinematic viscosity of the fluid.

The Reynolds number of the Couette flow Re_(c) is given by:

$\begin{matrix}{{Re}_{c} = \frac{U_{i}H}{v}} & (16)\end{matrix}$

where

-   -   U_(i) is the circumferential velocity of the inner cylinder 120,    -   H is the gap width of the Couette device (H=r₀−R), and    -   ν is the kinematic viscosity of the fluid.

Eq. (16) can be used to rewrite Eq. (15a) as follows:

$\begin{matrix}{D_{0^{*}} = {1 + {\alpha_{*}\frac{u_{0^{*}}{Re}_{c}}{2\; U_{i}}}}} & \left( {17a} \right)\end{matrix}$

Similarly, Eq. (16) can be used to rewrite Eq. (15b) as follows:

$\begin{matrix}{D_{i^{*}} = {1 + {\alpha_{*}\frac{u_{i^{*}}{Re}_{c}}{2\; U_{i}}}}} & \left( {17b} \right)\end{matrix}$

As described in Eqs. (11) and (12) above, the ratio

$\frac{u_{0^{*}}{Re}_{c}}{\; U_{i}}$of Eq. (17a) can be defined as:

$\begin{matrix}{\frac{u_{0^{*}}{Re}_{c}}{\; U_{i}} = {\left( {1 - \eta} \right)\sqrt{\frac{G}{2\;\pi}}}} & (18)\end{matrix}$

Eq. (18) can be used to rewrite Eq. (17a) as follows:

$\begin{matrix}{D_{0^{*}} = {1 + {\frac{\alpha_{*}}{2}\left( {1 - \eta} \right)\sqrt{\frac{G}{2\;\pi}}}}} & (19)\end{matrix}$

Similarly, the ratio

$\frac{u_{i^{*}}{Re}_{c}}{U_{i}}$of Eq. (17b) can be defined as:

$\frac{u_{0^{*}}{Re}_{c}}{U_{i}} = {\frac{\left( {1 - \eta} \right)}{\eta}\sqrt{\frac{G}{2\;\pi}}}$

Eq. (20) can be used to rewrite Eq. (17b) as follows:

$\begin{matrix}{D_{i^{*}} = {1 + {\frac{\alpha_{*}}{2}\frac{\left( {1 - \eta} \right)}{\eta}\sqrt{\frac{G}{2\;\pi}}}}} & (21)\end{matrix}$

By analogy with a pipe flow, the thickness δ₀ ⁺ of the viscous outersublayer 151 is related to the drag reduction parameter D_(0*) for theviscous outer sublayer 151 as follows:δ₀ ⁺=11.6D _(0*) ³  (22)

Similarly, the thickness δ_(i) ⁺ of the viscous inner sublayer 155 isrelated to the drag reduction parameter D_(i*) for the viscous innersublayer 155 as follows:δ_(i) ⁺=11.6D _(i*) ³  (23)

The corresponding dimensionless velocity λ₀ at the boundary of theviscous outer sublayer 151 can be given as:

$\begin{matrix}{\lambda_{0} = {\frac{\delta_{0}^{+}}{D_{0^{*}}} = {11.6\; D_{0^{*}}^{2}}}} & (24)\end{matrix}$

The corresponding dimensionless velocity λ_(i) at the boundary of theviscous inner sublayer 155 can be given as:

$\begin{matrix}{\lambda_{i} = {\frac{\delta_{i}^{+}}{D_{i^{*}}} = {11.6\; D_{i^{*}}^{2}}}} & (25)\end{matrix}$

The normalized velocity at the boundary of the inner viscous sublayer155 can be derived on the basis of Eq. (10) above to provide:

$\begin{matrix}{\frac{u\left( {r_{0} + \delta_{i}} \right)}{u_{0^{*}}} = {{\frac{u\left( R_{m} \right)}{u_{0^{*}}}\frac{2\;\eta}{1 + \eta}\left( {1 + m} \right)} + {\frac{1}{\kappa}\left( {{- \frac{1}{\eta}} + \frac{2\left( {1 + m} \right)}{1 + \eta} + {\frac{1 + m}{\eta}\ln\frac{\left( {1 - \eta} \right)}{\left( {1 + \eta} \right)}\frac{\left( {1 + m} \right)}{m}}} \right)}}} & (26)\end{matrix}$where m=δ_(i) ⁺/r₀ ⁺, r₀ ⁺=R⁺ where R⁺ is calculated by Eq. (7).

The normalized velocity at the gap centerline 154 of Eq. (26)

$\frac{u\left( R_{m} \right)}{u_{0^{*}}}$is given by Eqs. (9) and (6) as repeated below:

$\begin{matrix}{\frac{u\left( R_{m} \right)}{u_{0^{*}}} = {{\frac{1}{\kappa}\left( {1 + {\frac{1 + \eta}{2}{\ln\left( \frac{1 - \eta}{1 + \eta} \right)}}} \right)} + {\gamma\frac{1 + \eta}{2}}}} & (27) \\{{{{where}\mspace{14mu}\gamma} = \frac{\lambda - {\frac{1}{\kappa}\left( {1 + {\left( {1 - a} \right){\ln\left( \frac{a}{1 - a} \right)}}} \right)}}{1 - a}};} & (28)\end{matrix}$The parameter α=δ₀ ⁺/R⁺ and λ=λ₀ needed for Eq. (28) can be determinedfor the outer cylinder by Eqs. (19), (22) and (24).

The set of Eqs. (13), (19), (21)-(27) define a computational model forthe Couette flow that accounts for the drag reducing effects of the dragreducer. The computational model is dependent on the dimensions of theCouette device 100, including the radius R of the inner wall surface104A of the outer cylinder 104, the ratio η (which is the ratio r₀/R),and the height L of the gap of the Couette device 100. The parameterα_(*) is the major model variable that is a function of the drag reduceragent type and its concentration.

The fluid density ρ and the kinematic viscosity ν of the fluid aremeasured separately.

The shear stress τ_(w) and corresponding rotor angular velocity ω of theCouette device 100 are measured during operation of the Couette device100 for a given drag reducer agent type and its concentration.

The value of the parameter α_(*) for the given drag reducer agent typeand concentration of the test can be provided by statistical analysis ofexperimental data. Specifically, experiments can be carried out with theCouette device for a fluid solution that employs a given drag reducingadditive at a particular concentration where the shear stress τ_(w) ismeasured for a set of different rotor angular velocities ω. The set ofmeasurements of shear stress and corresponding rotor angular velocity aswell as the measured fluid density ρ and the kinematic viscosity ν areinput to the computation model based on Eqs. (13), (19), (21)-(27) tosolve for the parameter α_(*) for the given drag reducer agent type andits concentration.

An important aspect of drag reduction phenomenon, not accounted for bythe drag reduction model presented, is the maximum drag reductionasymptote. This asymptote provides the minimum Fanning friction factorsobtainable. The minimum friction factor obtainable in a pipe flow isdescribed by the empirical equation of Virk (1971):

$\begin{matrix}{\frac{1}{f^{0.5}} = {{19{\log_{10}\left( {{Re}\; f^{0.5}} \right)}} - 32.4}} & (29)\end{matrix}$

-   -   Equation (29) is applicable to a Couette flow, where the        Reynolds number (Re_(c)) number calculated by Eq. 16 is used        instead of Re and the Fanning friction factor is calculated for        the inner cylinder wall.

The Fanning friction factor f for the inner cylinder of the Couettedevice is calculated from the shear-stress equation applied to the innercylinder using Eqs. (1) and (7) and takes the following form:

$\begin{matrix}{\frac{1}{f^{0.5}} = \frac{{Re}_{c}}{\left( {\frac{1}{\eta} - 1} \right)\sqrt{\frac{G}{\pi}}}} & (30)\end{matrix}$

The preferable radius ratio η for the Couette device is below 0.7. Itfollows from the dependences 1/f^(0.5) versus Re f^(0.5) calculated fordifferent coefficients α_(*) ₀ at different η. The smaller the Couettedevice radius ratio, the stronger the curves 1/f^(0.5) vs. Re f^(0.5)are shifted downward from the drag reduction asymptote. The lower thesecurves are located, the wider range of the coefficient α_(*) ₀ can beidentified during testing different drag reducing chemicals.

C. Application of Couette Device Computational Model Solution of Part Bto Pipe Flow Modeling and Analysis

For a fluid solution employing a drag reducing agent that flows in apipe, the flow field for turbulent flow in the pipe can be described bytwo layers, which include a relatively viscous outer sublayer adjacentthe pipe wall and a turbulent inner core surrounded by the viscous outersublayer.

Applying the approach proposed by Yang and Dou in “Turbulent DragReduction with Polymer Additive in Rough Pipes,” Journal of FluidMechanics, Vol. 642, 2010, pp. 279-294, the velocity distribution acrossthe viscous sublayer at the pipe wall may be obtained in the followingform:u ⁺=2.5 ln y ⁺+11.6D _(*) ²−7.5 ln D _(*)−6.1  (31)

where

-   -   u⁺ is the normalized fluid velocity given by u⁺=u/u_(0*),    -   y⁺ given as y⁺=u_(*)y/ν where y is the distance from the pipe        wall, u_(*) is the friction velocity given by        u_(*)=(τ_(w)/ρ)^(0.5), τ_(w) is the shear stress at the pipe        wall, ρ is the density of the fluid of the solution, and ν is        the kinematic viscosity of the fluid of the solution; and    -   D_(*) is a drag reduction parameter.

The drag reduction parameter D_(*) of Eq. (31) is related to theparameter α_(*) that is a function of the drag reducer agent type andits concentration as described above in the computational model of PartB as follows:D _(*)=1+α_(*) R ⁺  (32)where R⁺ is the dimensionless pipe radius.

The normalized mean flow velocity

$\frac{U}{u_{*}}$can be calculated by averaging the velocity u⁺ of Eq. (31) over the pipecross-section as:

$\begin{matrix}{\frac{U}{u_{*}} = {\frac{1}{{\pi\left( R^{+} \right)}^{2}}{\int_{0}^{R}{u^{+}2\pi\;{R^{+}\left( {R^{+} - y^{+}} \right)}\ {\mathbb{d}y^{+}}}}}} & (33)\end{matrix}$

Furthermore, the normalized mean flow velocity

$\frac{U}{u_{*}}$is related to the friction factor f by:

$\begin{matrix}{\frac{U}{u_{*}} = \sqrt{\frac{2}{f}}} & (34)\end{matrix}$

-   -   where U is the mean flow velocity through the pipe, and        -   u_(*) is the friction velocity given by            u_(*)=(τ_(w)/ρ)^(0.5).            The friction factor f relates pressure loss due to friction            along a given length of the pipe to the mean flow velocity            through the pipe.

The integration of Eq. (33) can be performed analytically to obtain anequation for the friction factor f as follows:

$\begin{matrix}{\frac{1}{f^{0.5}} = {{4{\log_{10}\left( {{Re}\; f^{0.5}} \right)}} + {8.2D_{*}^{2}} - 8.6 - {12.2\log_{10}D_{*}}}} & (35)\end{matrix}$

As given by Eq. (32), the drag reduction parameter D_(*) of Eq. (35) isa function of the dimensionless pipe radius R⁺, which can be related tothe Reynolds number Re of the flow in the pipe and the friction factor fby:R ⁺=0.5Re√{square root over (f/2)}  (36)

The Reynolds number Re of Eq. (36) is Oven by:

$\begin{matrix}{{Re} = \frac{U\; D}{v}} & (37)\end{matrix}$

The dimensionless pipe radius is calculated as:

$\begin{matrix}{R^{+} = {{0.5\frac{u_{*}\; D}{v}} = {0.5\frac{\left( {\tau_{w}/\rho} \right)^{0.5}D}{v}}}} & (38)\end{matrix}$

-   -   where        -   u_(*) is the friction velocity given by            u_(*)=(τ_(w)/ρ)^(0.5),        -   D is the diameter of the pipe, and        -   ν is the kinematic viscosity of the fluid.

Equation (35) is the major model equation for calculating the frictionfactor f_(*) The friction factor can be used for engineeringcalculations of the pipe flow, such as the predicted pressure drop inthe pipe (over the length L) for the flow employing a particular dragreducer agent type and concentration, which is given as

${\Delta\; p} = {2\;\rho\; f\frac{U^{2}}{D}{L.}}$

The fluid density ρ and the kinematic viscosity ν of the fluid aremeasured separately.

The Reynolds number Re of the fluid flow is determined by the mean flowvelocity, the pipe diameter and the fluid kinematic viscosity, which areknown.

The parameter α_(*) that is a function of the drag reducer agent typeand its concentration is given by the solution of the computationalmodel in Part B for the given drag reducer agent type and itsconcentration.

These operations can be carried out for a number of differentconcentrations of a particular drag reducer agent type or over differentdrag reducer agents to characterize the expected pipe flow for thesedifferent scenarios. It can also be carried out for a number of fluidflows with different Reynolds number Re to characterize the expectedpipe flow for these different scenarios.

The computational models of Parts A, B, and C of the present applicationcan be realized by one or more computer programs (instructions and data)that are stored in the persistent memory (such as hard disk drive orsolid state drive) of a suitable data processing system and executed onthe data processing system. The data processing system can be a realizedby a computer (such as a personal computer or workstation) or a networkof computers.

Advantageously, the computational model of Part C characterizes the pipeflow of a dilute drag reducer polymer solution through the use of anempirical parameter that is a function of the drag reducer polymer typeand concentration. This empirical parameter can be derived from thesolution of a computation model for turbulent Couette flow of such dragreducer polymer solutions based upon experiments that generate andmeasure properties of the turbulent Couette flow for such dilute dragreducer polymer solutions as described in Part B above.

Furthermore, the computational models of the present application employa two layer representation of the boundary layer interfaces for both theturbulent Couette flow and the pipe flow in order to simplify theequations for such fluid flow. Specifically, the turbulent Couette flowis represented by three layers including viscous inner and outersublayers with a turbulent core therebetween, and the pipe flow isrepresented by a viscous outer sublayer that surrounds a turbulent core.The computation model of the turbulent Couette flow also provides forcomputation of the dimensionless torque applied to the Couette device asa function of the rotation speed for a given drag reducer polymer typeand its concentration.

There have been described and illustrated herein embodiments ofcomputational models that characterize the pipe flow of a dilute dragreducer polymer solution with the use of an empirical parameter that isa function of the drag reducer polymer type and its concentration. Thisempirical parameter can be derived from the solution of a computationmodel for turbulent Couette flow of such drag reducer polymer solutionsbased upon experiments that generate and measure properties of theturbulent Couette flow for such dilute drag reducer polymer solutions.While particular embodiments have been described, it is not intendedthat the embodiments be limited thereto. It will therefore beappreciated by those skilled in the art that yet other modificationscould be made to the provided embodiments without deviating from itsscope as claimed.

What is claimed is:
 1. A method of characterizing fluid flow in a pipewhere the fluid includes a drag reducing polymer of a particular typeand particular concentration, the method comprising: i) providing aCouette device that includes an annulus disposed between an outerannular surface and an inner annular surface and a sensor disposedadjacent the outer annular surface, wherein the sensor is configured tomeasure shear stress of fluid at the outer annular surface; ii)providing a first computational model associated with the Couette deviceof i) to model turbulent Couette flow of a fluid based upon a flowrepresentation consisting of an inner sublayer, an outer sublayer, and aturbulent core between the inner sublayer and the outer sublayer,wherein the first computational model includes an empirical parameterfor the particular type and the particular concentration of the dragreducing polymer that is based on the shear stress of the fluid at theouter annular surface measured by the sensor, and wherein the firstcomputational model further includes a first drag reduction parameterassociated with the outer sublayer and a second drag reduction parameterassociated with the inner sublayer, wherein both the first and seconddrag reduction parameters are based on the empirical parameter; iii)loading a fluid into the annulus of the Couette device of i), the fluidincluding the particular type and the particular concentration of thedrag reducing polymer, and operating the Couette device of i) to induceturbulent flow of the fluid in the annulus of the Couette device and toobtain related experimental data including shear stress of the fluid atthe outer annular surface measured by the sensor; iv) processing thefirst computational model of ii) in conjunction with the experimentaldata obtained in iii) on a data processor to solve for the empiricalparameter of the first computational model; v) providing a secondcomputational model that is configured to model flow of fluid in a pipe,wherein the second computational model includes a drag reductionparameter that is a function of the empirical parameter as solved for iniv), and wherein the second computational model further includes afriction factor that is a function of the drag reduction parameter,wherein the friction factor relates pressure loss due to friction alonga given length of pipe to the mean flow velocity through the pipe; andvi) processing the second computational model on a data processor toderive information that characterizes the flow of fluid in the pipe. 2.A method according to claim 1, wherein the drag reduction parameter isalso a function of a dimensionless pipe radius.
 3. A method according toclaim 2, wherein the second computational model is configured to relatethe drag reduction parameter to the empirical parameter by an equationof the form:D _(*)=1+α_(*) R ⁺ where D_(*) is the drag reduction parameter, α_(*) isthe empirical parameter, and R⁺ is the dimensionless pipe radius.
 4. Amethod according to claim 1, wherein the second computational model isconfigured to relate the friction factor to the drag reduction parameterby an equation of the form:$\frac{1}{f^{0.5}} = {{4{\log_{10}\left( {{Re}\; f^{0.5}} \right)}} + {8.2D_{*}^{2}} - 8.6 - {12.2\log_{10}D_{*}}}$where f is the friction factor, D_(*) is the drag reduction parameter,and Re is the Reynolds number of the flow in the pipe.
 5. A methodaccording to claim 4, wherein the second computational model is furtherconfigured to relate the Reynolds number Re to a dimensionless piperadius R⁺.
 6. A method according to claim 4, wherein the informationderived in iv) includes a solution for the friction factor f for givenflow conditions.
 7. A method according to claim 6, wherein theinformation derived in iv) includes a pressure drop over a given lengthof pipe based on the solution for the friction factor f.
 8. A methodaccording to claim 1, wherein the second computational model is basedupon a representation of the flow as two layers consisting of a viscousouter sublayer that surrounds a turbulent core.
 9. A method according toclaim 1, wherein both the first and second drag reduction parameters arealso based on a dimensionless torque G derived from the shear stress ofthe fluid at the outer annular surface measured by the sensor.
 10. Amethod according to claim 9, wherein: the outer and inner annularsurfaces of the Couette device of i) are concentric with respect to oneanother about a common center, wherein the outer annular surface isoffset from the center by a first radius R and the inner annular surfaceis offset from the center by a second radius r₀, wherein R is greaterthan r₀; and the first computational model is configured to relate thefirst and second drag reduction parameters to the empirical parameter byequations of the following form:$D_{o^{*}} = {1 + {\frac{\alpha_{*}}{2}\left( {1 - \eta} \right)\sqrt{\frac{G}{2\;\pi}}}}$$D_{i^{*}} = {1 + {\frac{\alpha_{*}}{2}\frac{\left( {1 - \eta} \right)}{\eta}\sqrt{\frac{G}{2\;\pi}}}}$where D_(0*) is the drag reduction parameter associated with the outersublayer, D_(i*) is the drag reduction parameter associated with theinner sublayer, α_(*) is the empirical parameter, η is the ratio r₀/R,and G is the dimensionless torque derived from the shear stress of thefluid at the outer annular surface measured by the sensor.
 11. A methodaccording to claim 1, wherein the first computational model includes anequation that defines a fluid velocity at a boundary of a viscoussublayer adjacent one of the outer and inner annular surfaces.
 12. Amethod according to claim 11, wherein the equation is derived bymomentum conservation for a portion of a turbulent core adjacent theviscous sublayer.
 13. A method according to claim 1, wherein: the sensorcomprises a floating element, a mechanical cantilever beam and a fiberBragg grating strain gauge.
 14. A method according to claim 9, wherein:the dimensionless torque G is determined based upon the followingrelation: $G = \frac{T}{\rho\; v^{2}L}$ where T is a torque derived froma shear stress τ_(w) as T=τ_(w)2πR²L, where the shear stress τ_(w) isthe shear stress of the fluid at the outer annular surface measured bythe sensor, R is the radius of the outer annular surface, ρ is thedensity of the fluid, ν is the kinematic viscosity of the fluid, and Lis the height of the gap of the Couette device.
 15. A method accordingto claim 1, wherein: the method is repeated for a number of differentconcentrations of a particular drag reducing polymer or over differentdrag reducing polymers to characterize the expected pipe flow for thesedifferent scenarios.
 16. A method according to claim 1, wherein: themethod is repeated for a number of different flow conditions tocharacterize the expected pipe flow for these different scenarios.
 17. Amethod of characterizing fluid flow in a pipe where the fluid includes adrag reducing polymer of a particular type and particular concentration,the method comprising: i) providing a Couette device that includes anannulus disposed between an outer annular surface and an inner annularsurface; ii) providing a first computational model associated with theCouette device of i) to model turbulent Couette flow of a fluid, whereinthe first computational model includes an empirical parameter for theparticular type and the particular concentration of the drag reducingpolymer; iii) loading a fluid into the annulus of the Couette device ofi), the fluid including the particular type and the particularconcentration of the drag reducing polymer, and operating the Couettedevice of i) to induce turbulent flow of the fluid in the annulus of theCouette device and to obtain related experimental data; iv) processingthe first computational model of ii) in conjunction with theexperimental data obtained in iii) on a data processor to solve for theempirical parameter; v) providing a second computational model that isconfigured to model flow of fluid in a pipe, wherein the secondcomputational model is configured to utilize the empirical parameter assolved for in iv); and vi) processing the second computational model ona data processor to derive information that characterizes the flow offluid in the pipe; wherein the first computational model includes afirst drag reduction parameter associated with the outer annular surfaceof the Couette device of i) and a second drag reduction parameterassociated with the inner annular surface Couette device of i), whereinboth the first and second drag reduction parameters are based on theempirical parameter and a dimensionless torque G applied to the Couettedevice; wherein the outer and inner annular surfaces of the Couettedevice of i) are concentric with respect to one another about a commoncenter, wherein the outer annular surface is offset from the center by afirst radius R and the inner annular surface is offset from the centerby a second radius r₀, wherein R is greater than r₀; and wherein thefirst computational model is configured to relate the first and seconddrag reduction parameters to the empirical parameter by equations of thefollowing form:$D_{o^{*}} = {1 + {\frac{\alpha_{*}}{2}\left( {1 - \eta} \right)\sqrt{\frac{G}{2\;\pi}}}}$$D_{i^{*}} = {1 + {\frac{\alpha_{*}}{2}\frac{\left( {1 - \eta} \right)}{\eta}\sqrt{\frac{G}{2\;\pi}}}}$where D_(0*) is the drag reduction parameter associated with the outerannular surface, D_(i*) is the drag reduction parameter associated withthe inner annular surface, α_(*) is the empirical parameter, η is theratio r₀/R, and G is the dimensionless torque applied to the Couettedevice.
 18. A method according to claim 17, wherein: the Couette deviceof i) includes a sensor disposed adjacent the outer annular surface,wherein the sensor is configured to measure shear stress of fluid at theouter annular surface; the empirical parameter of the first computationmodel is based on the shear stress of the fluid at the outer annularsurface measured by the sensor; and the experimental data obtained iniii) and processed in iv) includes the shear stress of the fluid at theouter annular surface measured by the sensor.
 19. A method according toclaim 18, wherein: the dimensionless torque G is determined based uponthe following relation ${G = \frac{T}{\rho\; v^{2}L}},$ where T is atorque derived from a shear stress τ_(w) as T=τ_(w)2πR²L, where theshear stress τ_(w) is the shear stress of the fluid at the outer annularsurface measured by the sensor, R is the radius of the outer annularsurface, L is the height of the gap of the Couette device, ρ is thedensity of the fluid, and ν is the kinematic viscosity of the fluid. 20.A method according to claim 1, wherein: the sensor is located in asensor enclosure disposed adjacent the outer annular surface.